# Clip 4/9: lesson part 1d

## Overview

In this clip, Mia Buljan works with her students on a number talk, using a mental math approach to multiplying a two-digit number and a one-digit number. She encourages her students to identify regularity among the various approaches identified, and guides them toward identifying the commutative property.

## Teacher Commentary

MIA BULJAN: I had collected the answers “69,” “70,” “56,” and “65” for the answers to the problem I presented (5 x14). Monique notices, during Enmy’s explanation of how she counted by five 14 times, that you can really count every two fives as a ten. Monique doesn’t know it, but she has invented the strategy of compensation in multiplication. She doubled one factor (5 became 10) and halved the other factor (14 became 7) so she turned 5 x 14 into 10 x 7.

This is not necessarily a strategy I’d expect my students to use yet, or even be able to articulate past this “special case” to generalizing about halving and doubling, or quartering and quadrupling, etc.

However, I did want to capture the idea so we could explore it as a conjecture (does it always work? Sometimes work? Never really work?) and to practice our productive discourse.

In the first part of this segment, as we wind down the strategy sharing, I see that I’m continuing to put more of the discussion focus on the students listening carefully to each other, rather than relying on me to revoice everything. I’m still doing quite a bit of the heavy lifting here, in terms of clarifying and orienting their thinking, but I am also giving them more space to practice hearing each other. When Enmy says she doesn’t understand Monique’s way, rather than revoicing or restating, I direct her to listen to Monique explain herself again.

In the final part of this segment, I ask them to use their whiteboards to write a multiplication number sentence. I like using whiteboards for this kind of experimental work. It’s less permanent, and therefore less intimidating, than pencil and paper work. In the video, a lot of kids are just swiping away their experiments in favor of other ideas as they discuss with their partners. There is still a lot of confusion on those whiteboards at the end of the number talk, but it was great formative information on where to go next. I imagine these students would benefit from articulating in words first, before modeling with their number sentences. Putting into words “5 equal groups of 10” to describe what Marlene had done is the conceptual part that goes with the more concrete number sentence.

Being able to connect these concepts is part of our Sociomathemical Norms. The answer of 70 is important, but it’s not the only math. Being able to connect our ideas, understand and question our classmates, and push our conceptual understanding of multiplicative reasoning is as important.